Station Two · Axiom 1 · Condition Skip to content

Station Two · Axiom 1

Condition — where symmetry is tested and proof first introduces motion.

👁 Condition Contrast → Distinction → Condition → Licence → Return → Truth Closed FOL

Axiom 1 — Condition

A condition acts on a contrasted pair and asks whether their relation holds. Proof introduces asymmetry — it pushes the pair one way under that condition. Where one side fits and the other does not, a licence opens. If that direction survives without return, the relation becomes a settled line.

Symmetry under test

Before proof, the pair ⟨👁A, 👁B⟩ is balanced. A condition steps in and asks whether the balance holds. Proof is what breaks that balance — a test that leans one way. Once the tilt appears, relation begins to move. The condition holds the rule of that movement.

Example of a condition

Take a simple rule: if the sun is not shining then the light is on. The condition tests a pair — weather and lamp. When the first term is false, the second must hold true. The line moves as 👁Sun ⇒ 👁Light. It is a one-way relation under a clear test of fit.

Proof and counter-falsification

A proof settles one direction. It says the condition holds and licences motion that way. A counter-falsification can strike it down, reversing the tilt. The relation then swings back to unresolved symmetry. Proof and falsification form the dynamic of movement around the condition.

Empirical reading

In observation, balance is fragile. Two identical twins taught in the same way may draw the same conclusion — we treat that with caution. It may be mere cultural symmetry. Yet when two researchers from different traditions or two contrasting experiments converge on the same result, confidence rises. Truth strengthens through contrastive convergence, not sameness.

Closure and return

Once a direction stands without contradiction, the licence closes into a return. We can write it as 👁A ⇒ 👁B. That closure marks coherence. It is no longer a free tilt but a resolved structure under the rule of its condition.

Station schema

Contrast (difference between relata) → Distinction (off the diagonal) → Condition (tests symmetry through proof) → Licence (one direction opens) → Counter-falsification (may reverse or remove the tilt) → Return (closure once challenge ceases) → Truth. Empirical trust grows when distinct points of view converge under one condition — contrastive coherence rather than sameness.

Axiom 1 · contrast → distinction → (condition + asymmetric licence) → return · Closed first-order form

Signature (closed FOL)

  • Predicates: D(x,y) contrast (irreflexive, symmetric), DistL(x) distinction-layer relatum, C(c,x,y) condition, L(c,x,y) licence, Ret(x,y) return, TAt(x), FAt(x) role labels (optional).
  • Extends Axiom 0 with the same symbols and optional role labels.

Base link carried from Axiom 0

∀x ∃y D(x,y)
∀x ¬D(x,x)
∀x ∀y ( D(x,y) ↔ x ≠ y )

Relational nature of condition

∀c ∀x ∀y ( C(c,x,y) → D(x,y) )

[A1.C0] Conditions live on contrasted pairs.

Licensing and unresolved opposition

  1. ∀c ∀x ∀y ( L(c,x,y) → C(c,x,y) )

    [A1.C1] Licence requires condition.

  2. ∀c ∀x ∀y ¬( L(c,x,y) ∧ L(c,y,x) )

    [A1.C2] No two-way licence under one condition.

  3. ∀x ∀y ( (∃c₁ L(c₁,x,y)) ∧ (∃c₂ L(c₂,y,x)) → D(x,y) ∧ ¬Ret(x,y) ∧ ¬Ret(y,x) )

    [A1.Opp] Opposition keeps the pair unresolved.

Return as uncontested licensed read

∀x ∀y ( Ret(x,y) ↔ ( ∃c ( C(c,x,y) ∧ L(c,x,y) ) ) ∧ ¬∃c' L(c',y,x) )

[A1.R≡] Exactly one direction licensed and no reverse licence.

Polarity of return (roles, if used)

∀x ∀y ( Ret(x,y) → TAt(x) ∧ FAt(y) )
∀x ¬( TAt(x) ∧ FAt(x) )

[A1.T0–T1] Optional role labels separate the poles of the return.

Unresolved without condition

∀x ∀y ( D(x,y) ∧ ¬∃c C(c,x,y) ∧ ¬∃c C(c,y,x) → ¬Ret(x,y) ∧ ¬Ret(y,x) )

[A1.U0] No condition means no return.

Resolution under uncontested condition

∀x ∀y ∀c ( D(x,y) ∧ C(c,x,y) ∧ L(c,x,y) ∧ ¬∃c' L(c',y,x) → Ret(x,y) ∧ TAt(x) ∧ FAt(y) )

[A1.R0] The uncontested licence settles the pair.

Directional exclusivity

∀x ∀y ¬( Ret(x,y) ∧ Ret(y,x) )

[A1.E0] There is no two-way return.

Derivations and checks

Theorem · Direction needs a condition

∀c ∀x ∀y ( L(c,x,y) → D(x,y) )

Use [A1.C1] to obtain L → C, then [A1.C0] to obtain C → D.

Theorem · No condition, no return

∀x ∀y ( ¬∃c C(c,x,y) ∧ ¬∃c C(c,y,x) → ¬Ret(x,y) ∧ ¬Ret(y,x) )

If there is no condition, the left conjunct of [A1.R≡] fails in both directions.

Example schema · negation as reverse licence

∀u ∀v ( Neg(u,v) → C(c_neg,v,u) ∧ L(c_neg,v,u) ∧ ¬∃c L(c,u,v) )

Then by [A1.R≡]:

∀u ∀v ( Neg(u,v) → Ret(v,u) ∧ TAt(v) ∧ FAt(u) )

Corollary · Return sits on the distinction layer

∀x ∀y ( Ret(x,y) → DistL(x) ∧ DistL(y) )

From [A1.R≡] pick a witness for C. Use [A1.C0] to get D, then apply Axiom 0’s layer clause.

Station schema

Contrast D → Distinction (x ≠ y) → Condition and LicenceReturn Ret. Opposition on a pair yields no return. Only an uncontested licence yields Ret.

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